{"about":{"site":"https://codewithpapers.app","non_affiliation":"Code with Papers and Syntology are not affiliated with, endorsed by, or sponsored by Papers with Code, Meta, or the pwc-archive mirror.","licence":"CC BY-SA 4.0","licence_url":"https://creativecommons.org/licenses/by-sa/4.0/legalcode","attribution":"https://codewithpapers.app/attribution","modified":"archive material modified by Syntology; see the attribution page"},"url":"/paper/the-sea-algorithm-for-endomorphisms-of","title":"The SEA algorithm for endomorphisms of supersingular elliptic curves","arxiv_id":"2501.16321","date":"2025-01-27","proceeding":null,"authors":["Travis Morrison","Lorenz Panny","Jana Sotáková","Michael Wills"],"abstract":"For a prime $p{\\,>\\,}3$ and a supersingular elliptic curve $E$ defined over $\\mathbb{F}_{p^2}$ with ${j(E)\\notin\\{0,1728\\}}$, consider an endomorphism $\\alpha$ of $E$ represented as a composition of $L$ isogenies of degree at most $d$. We prove that the trace of $\\alpha$ may be computed in $O(n^4(\\log n)^2 + dLn^3)$ bit operations, where $n{\\,=\\,}\\log(p)$, using a generalization of the SEA algorithm for computing the trace of the Frobenius endomorphism of an ordinary elliptic curve. When $L\\in O(\\log p)$ and $d\\in O(1)$, this complexity matches the heuristic complexity of the SEA algorithm. Our theorem is unconditional, unlike the complexity analysis of the SEA algorithm, since the kernel of an arbitrary isogeny of a supersingular elliptic curve is defined over an extension of constant degree, independent of $p$. We also provide practical speedups, including a fast algorithm to compute the trace of $\\alpha$ modulo $p$.","url_abs":"https://arxiv.org/abs/2501.16321v1","url_pdf":"https://arxiv.org/pdf/2501.16321v1.pdf","source":{"archive":"pwc-archive (Hugging Face), CC BY-SA 4.0","snapshot":"2025-07-28","licence_url":"https://creativecommons.org/licenses/by-sa/4.0/legalcode","row_kind":"links_only","authors_date_abstract":"arXiv metadata, CC0 1.0 (https://info.arxiv.org/help/license), from the Kaggle arXiv metadata snapshot of 2026-09-12"},"code_links":[{"paper_slug":"the-sea-algorithm-for-endomorphisms-of","repo_url":"https://github.com/travismo/beyond-the-sea","is_official":1,"mentioned_in_paper":1,"mentioned_in_github":0,"framework":"none","reach":null}],"tasks":[],"methods":[],"datasets_introduced":[],"methods_introduced":[],"results":[],"syntology":{"atlas_url":null,"mcp":null,"developers":"https://syntology.ai/developers"},"arxiv_metadata":null,"syntology_extracted_results":null}